Cross-Elasticities

A popular way to measure/express product interdependence, both in micro-economic theory as in econometrics and marketing research, is by the use of cross-elasticities. Cross-elasticities measure the effect of the change in the

marketing mix action of a particular product/category on the sales of other products/categories.

3.2.1.1 Cross-price elasticity

Most of the work on the measurement of (and causes for) interdependence originates from micro-economic theory. In fact, the theoretical foundations of the problem of product purchase interdependencies were already studied in the beginning of the 20^th century in the context of the micro-economic models of price elasticity [214, 238]. Especially, the work of Triffin about cross-price elasticity can be considered as a significant contribution to the field.

Triffin’s microeconomic theory of cross-price elasticity [269] dates back to 1940 and defines complementarity (resp. substitution) between two products X and Y, whenever a price decrease (increase) of product X, i.e. (∆pX), generates higher (resp. lower) sales (∆SY) for Y. Cross-price elasticity in this context is therefore measured by the cross-price elasticity coefficient εXY :

(3.1)

A positive value for εXY indicates a substitution effect between X and Y, whereas a negative value indicates a complementary effect. Although elegant in its simplicity, an important limitation, namely the huge effort to measure all elasticities for a typically wide product assortment, has made implementations within a supermarket environment practically infeasible [200]. More recently, Blattberg and Neslin [35] developed a model for maximizing the profits in a category, taking into account interdependencies between items in the category.

The sales of each item are made dependent on the other items’ deals. As a result, the category margin and the degree of cannibalization determine the optimal price discount for an item. Also Mulhern and Leone [209] studied the

Y Y

X X

XY

SS p p

ε

-50-purchase complements, and the item’s -50-purchase substitutes. They found that a price deal increases the sales volume of the promoted item and the item’s purchase complements, but reduces the sales of the item’s purchase substitutes.

3.2.1.2 Cross-space elasticity

Later, the concept of cross-elasticities was used by marketeers in the context of shelf-space allocation models to express the impact of shelf-space decisions of one product/category on the sales of other products/categories. For instance, Corstjens and Doyle [83] were the first to include both direct and cross-space elasticities into their space optimization model. They argue that any shelf-space allocation model to optimize a retailer’s profits should take into account both direct and cross-space elasticities. Later, cross-space elasticities were also adopted by Borin, Farris and Freeland [44], Urban [273] and Bultez et al. [65, 66] and Swinnen [257].

Unfortunately, obtaining good estimates of direct and cross-space elasticities for large amounts of products is not straightforward [174]. The literature describes three techniques in this context: experiments, time-series data and cross-sectional data. In-store experiments are probably the most reliable since they experimentally measure the effect of a change in shelf space on the sales.

However, since these experiments are very laborious, time consuming and may even be disruptive towards the operation of the store, this method is not used very often. Therefore, Bultez et al. [65, 66] used time-series data to estimate the effect of changing shelf space on sales. Finally, cross-sectional data offers an alternative solution to the measurement problem of elasticities [83, 84].

The idea is that when collecting data from different stores, there is enough variation in the amount of shelf space devoted to products and their resulting sales such that their relation can be estimated by regression techniques. The advantages are speedier results, low cost and no interference with store operations. However, the major drawback is the problem of identification since it is not always clear whether the relation between space and sales is the result

of true space elasticity, or whether it merely reflects the retailer’s earlier decisions to allocate more space in proportion to past or expected sales.

The above mentioned problems, and the computational difficulties as a result of the non-linear character of the optimization, may explain why many shelf space allocation models do not adopt elasticities [131] or when they do, they usually only consider a very limited number of products/categories.

3.2.1.3 Cross-location elasticity

Also the location of products within the store may have an impact on sales. For instance, Drèze et al. [99] discuss how retailers can boost sales by better managing their available shelf-space through reorganizing the location of the existing products in the assortment. In this context, eye-level is often seen as the best location. It is therefore crucial to carefully think about which products to put at those locations and how the reorganization may affect the sales of other products. By means of experiments, they found out that indeed the location of products has an important effect on the sales of the product and on related products. For instance, they showed that by putting toothbrush at eye-level, instead of toothpaste, the sales of toothbrushes increased by 8% whilst keeping toothpaste sales unaffected.

Chapter 5 in this dissertation is devoted to a discussion of our own optimization model to support such location decisions. The idea is that retailers often put top-selling products at visually attractive locations but that their rule-of-thumb usually does not take into account cross-selling effects with other products. In other words, even though a product is not a top-seller, but belongs to the sub-top selling group, its cross-selling effects with other products may be significant such that overall it can compete with (or exceed) some top-selling products and should therefore deserve an opportunity to be located at an attractive location.